S 1 Supporting Information for : Apparent Power-Law Behavior of Water ’ s Isothermal Compressibility and Correlation Length upon Supercooling

The isothermal compressibility and correlation length of supercooled water obtained from small-angle X-ray scattering (SAXS) were analyzed by fits based on an apparent power-law in the temperature range from 280 K down to the temperature of maximum compressibility at 229 K. Although the increase in thermodynamic response functions is not towards a critical point, it is still possible to obtain an apparent power law all the way to the maximum values with best-fit exponents of γ = 0.40 ± 0.01 for the isothermal compressibility and ν = 0.26 ± 0.03 for the correlation length. The ratio between these exponents is close to a value of ≈0.5, as expected for a critical point, indicating the proximity of a potential second critical point. Comparison of γ obtained from experiment with molecular dynamics simulations on the iAMOEBA water model shows that it would be located at pressures in the neighborhood of 1 kbar. The high value and sharpness of the compressibility maximum observed in the experiment are not reproduced by any of the existing classical water models, thus inviting further development of simulation models of water.


I.
Temperature validation i) Comparison to homogeneous nucleation rates: Assuming that the isothermal compressibility (T) would follow the apparent power law that is given by Speedy and Angell 1 which is shown as black dashed line in Figure 2(a) of the main script, we would expect the lowest temperature that was measured in the experiment to be ≈234 K instead of the reported 227.7 K.We estimate the probability of a homogeneous nucleation event to occur inside a droplet before it is hit by an x-ray pulse by where  hom is the number of droplets which froze homogeneously,  is the total number of solid and liquid droplets, V is the volume of a water droplet,  hom (()) = exp(−2.92()+ 706.5)cm -3 s -1 is the temperature-dependent homogenous nucleation rate reported by Murray et al. 2 , () is the time-dependent droplet temperature and  max is the lifetime of the droplet before it is hit by an x-ray pulse.The integral in the exponent is due to the droplet temperature changing during evaporation.We note that the change in droplet volume due to evaporation is much smaller than the experimentally observed uncertainty in the droplet diameter and its effect on the nucleation probability is neglected.
For the calculation, we assumed the droplet temperature to follow the temperature profile that is obtained by Knudsen theory of evaporation but rescaled to the hypothetically lowest temperature of 234 K.For the experimental condition of a 14 m droplet with a lifetime of 4 ms, we obtain a nucleation probability of ≈0.15%.This means that we would expect Bragg peaks from ice in only 1 out of ≈650 recorded scattering patterns in case the lowest temperature would be 234 K.This is in conflict with the experimental observations where almost all droplets were frozen at the lowest temperature.Anomalous components of T (circles) obtained from the experimental T after subtraction of the normal components in (a) and corresponding power-law fits according to eq. ( 1) in the main text (red and blue solid lines).
Microscopically, T is directly related to the volume fluctuations (V) in the liquid by 〈() 2 〉 =     .The influence on T of these fluctuations can be divided into a normal and an anomalous contribution, even though there is no strict division based on thermodynamic grounds.The normal component is related to thermally induced fluctuations that exist in any liquid and decreases with decreasing temperature, whereas the anomalous component of T is attributed to structural changes between HDL and LDL and therefore increases with decreasing temperature 8,9 .Figure S2(a) shows the experimental T and its minimum around 319 K. Usually, a normal component is subtracted from T in order to obtain a monotonic decrease with temperature in the remaining anomalous part of T, which then can be fitted by a power law.
In other studies 6,7 , the normal component for T has been approximated based on the influence of small impurities in the liquid.Here we estimate the normal and anomalous components directly from the fitting of the SAXS structure factor, that is divided into a normal contribution from a hard-sphere fluid and an anomalous contribution described by S5 Ornstein-Zernike theory.For detailed description of the fitting of the SAXS structure factor, see Refs.[3,10].
The decomposition of  T into normal and anomalous contributions is an approximation and depends on the approach that is used, as shown in Figure S2.As a consequence, also the apparent power-law fits and corresponding critical exponents differ for the different approaches.We find a large variance in the critical exponent of  = 1.3 and  = 2.1 for the anomalous components after decomposition suggested by Refs.[6,7] and decomposition using the SAXS structure factor, respectively.Therefore, we decided not to decompose T into a normal and anomalous contribution and instead use a limited temperature range for the power-law analysis, where T shows a monotonous decrease with temperature.We found reliable power-law fits of T for temperatures up to 300 K which supports the use of the entire experimental data set of Ref. [3], which was measured up to 281 K.

3 ii)
Comparison to droplet sizes measured by Raman resonance peaks: A recent study byGoy et al.4 has validated the applicability of Knudsen theory of evaporative cooling to supercooled water microdroplets, although claiming that the temperatures in a previous study using x-ray laser in a similar fashion as in the current study may be underestimated.5This is surprising since both the studies use the same experimental set-up and the same theory of Knudsen evaporation.Their procedure involves the extraction of the droplet diameter from measurements of the Raman resonance peaks and fitting them by an S3 iterative application of the Knudsen theory of evaporative cooling as a function of distance, using the droplet velocity and initial droplet diameter as fit parameters.Their best-fit is shown in FigureS1with an estimated droplet velocity of 22.2±1.5 m/s.The 68% confidence interval (corresponding to ± standard deviation from the mean value) of their fit corresponds to the shaded region as shown in FigureS1.Our proposed fit would have a lower velocity of 15.4 m/s, which would be consistent with the temperature estimation of the current study.This low value of velocity also fits the data much better at measured diameter values during the start of the evaporative cooling.This region corresponds to rapid evaporative cooling and sharp changes in droplet diameters and is thus the most sensitive part for the fitting procedure as compared to later stages of evaporative cooling.At longer times, there can be expansion of the droplets due to the presence of nanometer ice crystals that may not be detectable in the Raman measurements when the droplet is not yet fully frozen.

Figure
Figure S1. Figure adapted from Goy et al. 4 showing their measurement, fit to their measurement with the shaded region corresponding to ± standard deviation from the mean of the fitting parameters.The red line corresponds to our proposed value for the velocity.